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G = C4.S4order 96 = 25·3

2nd non-split extension by C4 of S4 acting via S4/A4=C2

Aliases: C4.2S4, Q8.3D6, CSU2(𝔽3)⋊2C2, SL2(𝔽3).3C22, C2.8(C2×S4), C4○D4.2S3, C4.A4.1C2, SmallGroup(96,191)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C4.S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — CSU2(𝔽3) — C4.S4
 Lower central SL2(𝔽3) — C4.S4
 Upper central C1 — C2 — C4

Generators and relations for C4.S4
G = < a,b,c,d,e | a4=d3=1, b2=c2=e2=a2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a2b, dbd-1=a2bc, ebe-1=bc, dcd-1=b, ece-1=a2c, ede-1=d-1 >

Character table of C4.S4

 class 1 2A 2B 3 4A 4B 4C 4D 6 8A 8B 12A 12B size 1 1 6 8 2 6 12 12 8 12 12 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ5 2 2 2 -1 2 2 0 0 -1 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 -1 -2 2 0 0 -1 0 0 1 1 orthogonal lifted from D6 ρ7 3 3 1 0 -3 -1 1 -1 0 1 -1 0 0 orthogonal lifted from C2×S4 ρ8 3 3 1 0 -3 -1 -1 1 0 -1 1 0 0 orthogonal lifted from C2×S4 ρ9 3 3 -1 0 3 -1 -1 -1 0 1 1 0 0 orthogonal lifted from S4 ρ10 3 3 -1 0 3 -1 1 1 0 -1 -1 0 0 orthogonal lifted from S4 ρ11 4 -4 0 -2 0 0 0 0 2 0 0 0 0 symplectic faithful, Schur index 2 ρ12 4 -4 0 1 0 0 0 0 -1 0 0 √3 -√3 symplectic faithful, Schur index 2 ρ13 4 -4 0 1 0 0 0 0 -1 0 0 -√3 √3 symplectic faithful, Schur index 2

Smallest permutation representation of C4.S4
On 32 points
Generators in S32
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 6 3 8)(2 7 4 5)(9 26 11 28)(10 27 12 25)(13 18 15 20)(14 19 16 17)(21 32 23 30)(22 29 24 31)
(1 18 3 20)(2 19 4 17)(5 16 7 14)(6 13 8 15)(9 22 11 24)(10 23 12 21)(25 30 27 32)(26 31 28 29)
(5 17 14)(6 18 15)(7 19 16)(8 20 13)(9 26 31)(10 27 32)(11 28 29)(12 25 30)
(1 22 3 24)(2 21 4 23)(5 27 7 25)(6 26 8 28)(9 20 11 18)(10 19 12 17)(13 29 15 31)(14 32 16 30)```

`G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,6,3,8)(2,7,4,5)(9,26,11,28)(10,27,12,25)(13,18,15,20)(14,19,16,17)(21,32,23,30)(22,29,24,31), (1,18,3,20)(2,19,4,17)(5,16,7,14)(6,13,8,15)(9,22,11,24)(10,23,12,21)(25,30,27,32)(26,31,28,29), (5,17,14)(6,18,15)(7,19,16)(8,20,13)(9,26,31)(10,27,32)(11,28,29)(12,25,30), (1,22,3,24)(2,21,4,23)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30)>;`

`G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,6,3,8)(2,7,4,5)(9,26,11,28)(10,27,12,25)(13,18,15,20)(14,19,16,17)(21,32,23,30)(22,29,24,31), (1,18,3,20)(2,19,4,17)(5,16,7,14)(6,13,8,15)(9,22,11,24)(10,23,12,21)(25,30,27,32)(26,31,28,29), (5,17,14)(6,18,15)(7,19,16)(8,20,13)(9,26,31)(10,27,32)(11,28,29)(12,25,30), (1,22,3,24)(2,21,4,23)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30) );`

`G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,6,3,8),(2,7,4,5),(9,26,11,28),(10,27,12,25),(13,18,15,20),(14,19,16,17),(21,32,23,30),(22,29,24,31)], [(1,18,3,20),(2,19,4,17),(5,16,7,14),(6,13,8,15),(9,22,11,24),(10,23,12,21),(25,30,27,32),(26,31,28,29)], [(5,17,14),(6,18,15),(7,19,16),(8,20,13),(9,26,31),(10,27,32),(11,28,29),(12,25,30)], [(1,22,3,24),(2,21,4,23),(5,27,7,25),(6,26,8,28),(9,20,11,18),(10,19,12,17),(13,29,15,31),(14,32,16,30)])`

C4.S4 is a maximal subgroup of
C8.S4  C8.4S4  Q8.4S4  D4.S4  GL2(𝔽3)⋊C22  Q8.6S4  D4.5S4  CSU2(𝔽3)⋊S3  C12.6S4  C4.S5  CSU2(𝔽3)⋊D5  C20.2S4
C4.S4 is a maximal quotient of
Q8⋊Dic6  CSU2(𝔽3)⋊C4  Q8.D12  SL2(𝔽3).D4  (C2×C4).S4  C12.3S4  CSU2(𝔽3)⋊S3  C12.6S4  CSU2(𝔽3)⋊D5  C20.2S4

Matrix representation of C4.S4 in GL4(𝔽3) generated by

 0 2 1 1 1 1 2 0 1 1 0 1 2 0 2 2
,
 2 1 1 1 0 2 2 1 2 2 0 1 2 0 2 2
,
 2 2 2 2 0 1 0 1 2 2 1 1 0 1 0 2
,
 2 2 2 2 1 2 1 2 2 2 0 1 1 2 2 0
,
 0 0 2 0 0 0 0 2 1 0 0 0 0 1 0 0
`G:=sub<GL(4,GF(3))| [0,1,1,2,2,1,1,0,1,2,0,2,1,0,1,2],[2,0,2,2,1,2,2,0,1,2,0,2,1,1,1,2],[2,0,2,0,2,1,2,1,2,0,1,0,2,1,1,2],[2,1,2,1,2,2,2,2,2,1,0,2,2,2,1,0],[0,0,1,0,0,0,0,1,2,0,0,0,0,2,0,0] >;`

C4.S4 in GAP, Magma, Sage, TeX

`C_4.S_4`
`% in TeX`

`G:=Group("C4.S4");`
`// GroupNames label`

`G:=SmallGroup(96,191);`
`// by ID`

`G=gap.SmallGroup(96,191);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,2,-2,288,601,295,146,579,447,117,364,286,202,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=d^3=1,b^2=c^2=e^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^2*c,e*d*e^-1=d^-1>;`
`// generators/relations`

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